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Advanced mathematics
This article is College level Direct product From Wikipedia:Direct product In , one can often define of objects already known, giving a new one. This of the underlying ,}} together with a suitably defined structure on the product set. More abstractly, one talks about the , which formalizes these notions. Examples are the product of sets, (described below), , and other . The of s is another instance. There is also the – in some areas this is used interchangeably, while in others it is a different concept. Examples * If we think of \mathbb{R} as the set of real numbers, then the direct product \mathbb{R}\times \mathbb{R} is just the Cartesian product \{ (x,y) \mid x,y \in \mathbb{R} \} . * If we think of \mathbb{R} as the of real numbers under addition, then the direct product \mathbb{R}\times \mathbb{R} still has \{ (x,y) \mid x,y \in \mathbb{R} \} as its underlying set. The difference between this and the preceding example is that \mathbb{R}\times \mathbb{R} is now a group, and so we have to also say how to add their elements. This is done by defining (a,b) + (c,d) = (a+c, b+d) . * If we think of \mathbb{R} as the of real numbers, then the direct product \mathbb{R}\times \mathbb{R} again has \{ (x,y) \mid x,y \in \mathbb{R} \} as its underlying set. The ring structure ring consists of addition defined by (a,b) + (c,d) = (a+c, b+d) and multiplication defined by (a,b) (c,d) = (ac, bd) . * However, if we think of \mathbb{R} as the of real numbers, then the direct product \mathbb{R}\times \mathbb{R} does not exist – naively defining addition and multiplication componentwise as in the above example would not result in a field since the element (1,0) does not have a . In a similar manner, we can talk about the direct product of finitely many algebraic structures, e.g. \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} . This relies on the fact that the direct product is up to . That is, (A \times B) \times C \cong A \times (B \times C) for any algebraic structures A , B , and C of the same kind. The direct sum is also up to isomorphism, i.e. A \times B \cong B \times A for any algebraic structures A and B of the same kind. We can even talk about the direct product of infinitely many algebraic structures; for example we can take the direct product of many copies of \mathbb R , which we write as \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \dotsb . Group direct product In one can define the direct product of two groups (G'', ∘) and (''H, ∙), denoted by G'' × ''H. For s which are written additively, it may also be called the , denoted by G \oplus H . It is defined as follows: * the of the elements of the new group is the Cartesian product of the sets of elements of G'' and ''H, that is {(g'', ''h): g'' ∈ ''G, h'' ∈ ''H}; * on these elements put an operation, defined element-wise: *: (g'', ''h) × (g , h' '') = (''g ∘ g , h'' ∙ ''h ) (Note that (G'', ∘) may be the same as (''H, ∙)) This construction gives a new group. It has a isomorphic to G'' (given by the elements of the form (''g, 1)), and one isomorphic to H'' (comprising the elements (1, ''h)). The reverse also holds, there is the following recognition theorem: If a group K'' contains two normal subgroups ''G and H'', such that ''K= GH and the intersection of G'' and ''H contains only the identity, then K'' is isomorphic to ''G × H''. A relaxation of these conditions, requiring only one subgroup to be normal, gives the . As an example, take as ''G and H'' two copies of the unique (up to isomorphisms) group of order 2, ''C''2: say {1, ''a} and {1, b''}. Then ''C''2×''C''2 = {(1,1), (1,''b), (a'',1), (''a,b'')}, with the operation element by element. For instance, (1,''b)*(a'',1) = (1*''a, b''*1) = (''a,b''), and (1,''b)*(1,b'') = (1,''b''2) = (1,1). With a direct product, we get some natural s for free: the projection maps define by : \begin{align} \pi_1: G \times H \to G, \ \ \pi_1(g, h) &= g \\ \pi_2: G \times H \to H, \ \ \pi_2(g, h) &= h \end{align} called the '''coordinate functions'. Also, every homomorphism f'' to the direct product is totally determined by its component functions f_i = \pi_i \circ f . For any group (''G, ∘) and any integer n'' ≥ 0, repeated application of the direct product gives the group of all ''n- s G''n'' (for n'' = 0 we get the ), for example '''Zn'' and '''Rn''. Direct product of modules The direct product for (not to be confused with the ) is very similar to the one defined for groups above, using the Cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components. Starting from '''R' we get R''n'', the prototypical example of a real n''-dimensional vector space. The direct product of '''Rm'' and '''Rn'' is '''Rm''+''n''. \bigoplus_{i=1}^n X_i . The direct sum and direct product differ only for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries.}} They are dual in the sense of : the direct sum is the , while the direct product is the product. Direct sum From Wikipedia:Direct sum The '''direct sum is an operation from , a branch of . For example, the direct sum \mathbf{R} \oplus \mathbf{R} , where \mathbf{R} is , is the , \mathbf{R} ^2 . To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the . The direct sum of two s A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B . *(Confusingly this ordered pair is also called the of the two groups.) To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d) ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as , , and s. *This relies on the fact that the direct sum is up to . That is, (A \oplus B) \oplus C \cong A \oplus (B \oplus C) for any algebraic structures A , B , and C of the same kind. *The direct sum is also up to isomorphism, i.e. A \oplus B \cong B \oplus A for any algebraic structures A and B of the same kind. }}. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression xy ) we use direct product. In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there would be a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. *More generally, if a + sign is used, all but finitely many coordinates must be zero, while *if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are (A_i)_{i \in I} , the direct sum \bigoplus_{i \in I} A_i is defined to be the set of tuples (a_i)_{i \in I} with a_i \in A_i such that a_i=0 for all but finitely many i''. The direct sum \bigoplus_{i \in I} A_i is contained in the \prod_{i \in I} A_i , but is usually strictly smaller when the I is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero. Spaces From Wikipedia:Topological space: Around 1735, Euler discovered the formula V - E + F = 2 relating the number of vertices, edges and faces of a convex polyhedron, and hence of a . No metric is required to prove this formula. The study and generalization of this formula is the origin of . may be defined as a set of points, along with a set of neighbourhoods for each point,}} satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. is a function that defines a concept of distance between any two points. The distance from a point to itself is zero. The distance between two distinct points is positive.}} : From Wikipedia:Normed vector space A norm is the generalization to real vector spaces of the intuitive notion of distance in the real world. All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same). From Wikipedia:Norm (mathematics) is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero.}} # \|\mathbf{v}\| ≥ 0 # \|\mathbf{v}\| = 0 \quad \quad \mathbf{v} = \mathbf{0} (the ) # \|\mathbf{u} + \mathbf{v}\| ≤ \|\mathbf{u}\| + \|\mathbf{v}\| \quad (The ) # \|\mathbf{v}\| = \|\mathbf{-v}\| # A , on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). Inner product From Wikipedia:Inner product space In the folowing, the of s denoted is either the field of s or the field of s . together with an inner product}}, i.e., with a map : \langle \cdot, \cdot \rangle : V \times V \to F that satisfies the following three properties for all vectors and all scalars : * symmetry: *: \langle x, y \rangle = \overline{\langle y, x \rangle} * in the first argument: *: \begin{align} \langle ax, y \rangle &= a \langle x, y \rangle \\ \langle x + y, z \rangle &= \langle x, z \rangle + \langle y, z \rangle \end{align} * : *: \langle x, x \rangle > 0,\quad x \in V \setminus \{\mathbf{0}\}. Positive-definiteness and linearity, respectively, ensure that: : \begin{align} \langle x, x \rangle &= 0 \Rightarrow x = \mathbf{0} \\ \langle \mathbf{0}, \mathbf{0} \rangle &= \langle 0x, 0x \rangle = 0 \langle x, 0x \rangle = 0 \end{align} Notice that conjugate symmetry implies that }} is real for all , since we have: : \langle x, x \rangle = \overline{\langle x, x \rangle} \,. Conjugate symmetry and linearity in the first variable imply : \begin{align} \langle x, a y \rangle &= \overline{\langle a y, x \rangle} = \overline{a} \overline{\langle y, x \rangle} = \overline{a} \langle x, y \rangle \\ \langle x, y + z \rangle &= \overline{\langle y + z, x \rangle} = \overline{\langle y, x \rangle} + \overline{\langle z, x \rangle} = \langle x, y \rangle + \langle x, z \rangle \,; \end{align} that is, in the second argument. So, . Conjugate symmetry is also called Hermitian symmetry, and a conjugate-symmetric sesquilinear form is called a Hermitian form. While the above axioms are more mathematically economical, This important generalization of the familiar square expansion follows: : \langle x + y, x + y \rangle = \langle x, x \rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle \,. These properties, constituents of the above linearity in the first and second argument: : \begin{align} \langle x + y, z \rangle &= \langle x, z \rangle + \langle y, z \rangle \,, \\ \langle x, y + z \rangle &= \langle x, y\rangle + \langle x, z \rangle \end{align} are otherwise known as . In the case of R'}}, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. So, That is, : \begin{align} \langle x, y \rangle &= \langle y, x \rangle \\ \Rightarrow \langle -x, x \rangle &= \langle x, -x \rangle \,, \end{align} and the becomes: : \langle x + y, x + y \rangle = \langle x, x \rangle + 2\langle x, y \rangle + \langle y, y \rangle \,. A common special case of the inner product, the scalar product or , is written with a centered dot a \cdot b . Bra-ket and , prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first.}} Then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product }} as | x''}}}} (the of ), respectively (dot product as a case of the convention of forming the matrix product as the dot products of rows of with columns of ). Here the kets and columns are identified with the vectors of and the bras and rows with the s (covectors) of the }}, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature, taking }} to be conjugate linear in rather than . A few instead find a middle ground by recognizing both }} and | ·}}}} as distinct notations differing only in which argument is conjugate linear. There are various technical reasons why it is necessary to restrict the to and in the definition. Briefly, the basefield has to contain an in order for non-negativity to make sense, and therefore has to have equal to 0 (since any ordered field has to have such characteristic). This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of or will suffice for this purpose, e.g., the s or the s. However, in these cases when it is a proper subfield (i.e., neither nor ) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over or , such as those used in , are automatically and hence s. List of Spaces From Wikipedia:List of vector spaces in mathematics: * ** ** * ** ** ** *** *** **** ***** ***** ****** ****** ****** ****** ******* ******** ********* ********* ********** ******* * * * * ** * * * Back to top Commutator In , fails to be .}} There are different definitions used in and . (including any ) is defined by}} : In , if two s of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. , every associative algebra can be turned into a .}} The anticommutator of two elements and of a ring or an associative algebra is defined by : \{a, b\} = ab + ba. Sometimes a,b_+ is used to denote anticommutator, while a,b_- is then used for commutator. The anticommutator is used less often, but can be used to define s and s, and in the derivation of the in particle physics. The commutator of two operators acting on a is a central concept in , since it quantifies how well the two s described by these operators can be measured simultaneously. The is ultimately a theorem about such commutators, by virtue of the . In , equivalent commutators of function are called s, and are completely isomorphic to the Hilbert-space commutator structures mentioned. The commutator has the following properties: Lie-algebra identities # + B, C = C + C # A = 0 # B = -A # [B, C] + [C, A] + [A, B] = 0 Relation (3) is called , while (4) is the . Additional identities # BC = BC + BC # BCD = BCD + BCD + BCD # BCDE = BCDE + BCDE + BCDE + BCDE # C = AC + CB # D = ABD + ADC + DBC # E = ABCE + ABED + AECD + EBCD # B + C = B + C # + B, C + D = C + D + C + D # CD = ACD + CBD + CAD + CDB # [C, D] = [[[B, C], D] + [[[C, D], A] + [[[D, A], B] + [[[A, B], C] If is a fixed element of a ring R'', identity (1) can be interpreted as a for the map \operatorname{ad}_A: R \rightarrow R given by \operatorname{ad}_A(B) = B . In other words, the map ad''A defines a on the ring R''. Identities (2), (3) represent Leibniz rules for more than two factors, and are valid for any derivation. Identities (4)–(6) can also be interpreted as Leibniz rules. Identities (7), (8) express '''Z'- . Some of the above identities can be extended to the anticommutator using the above ± subscript notation. For example: # C_- = AC_\mp \pm C_\mp B # CD_- = AC_\mp D \pm ACD_\mp + C_\mp DB \pm CD_\mp B # \left[B, C_\pm\right] + \left[C, A_\pm\right] + \left[A, B_\pm\right] = 0 Linear groups of order n is an n-by-n matrix. Any two square matrices of the same order can be added and multiplied. A matrix is invertible if and only if its determinant is nonzero.}} [[Wikipedia:General linear group|GL''n''(F'')]] or , or simply GL(''n) is the of invertible matrices with entries from the field F''. The group operation is . The group GL(n, F) and its subgroups are often called linear groups or matrix groups. :[[Wikipedia:Special linear group|SL(''n, F'')]] or SL''n(F''), is the of consisting of matrices with a of 1. :[[Wikipedia:Unitary group|U(''n)]], the Unitary group of degree n'' is the of . The group operation is .Wikipedia:Special unitary group The determinant of a unitary matrix is a complex number with norm 1. ::[[Wikipedia:special unitary group|SU(''n)]], the special unitary group of degree , is the of with 1. Back to top Symmetry groups : : boosts, rotations, translations :: : boosts, rotations :::The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.) Aff(n,K): the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. :E(n): rotations, reflections, and translations. ::O(n): rotations, reflections ::: : rotations}} :::: and consists of all matrices.}} Clifford group: The set of invertible elements ''x such that for all v'' in ''V x v \alpha(x)^{-1}\in V . The Q'' is defined on the Clifford group by Q(x) = x^\mathrm{t}x. :[[Wikipedia:Pin group|Pin''V(K'')]]: The subgroup of elements of spinor norm 1. Maps 2-to-1 to the orthogonal group ::[[Wikipedia:Spin group|Spin''V(K'')]]: The subgroup of elements of Dickson invariant 0 in Pin''V(K''). When the characteristic is not 2, these are the elements of determinant 1. Maps 2-to-1 to the special orthogonal group. Elements of the spin group act as linear transformations on the space of spinors Back to top Rotations :See also: , , , , , , , , , , , , :From Wikipedia:Rotation group SO(3): Consider the solid ball in '''R'3 of radius π. For every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The two rotations through π and through −π are the same. So we (or "glue together") on the surface of the ball. The ball with antipodal surface points identified is a , and this manifold is to the rotation group. It is also diffeomorphic to the RP'3, so the latter can also serve as a topological model for the rotation group. These identifications illustrate that SO(3) is but not . As to the latter, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that φ runs from 0 to 4π, you get a closed loop which ''can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. and similar tricks demonstrate this practically.}} The same argument can be performed in general, and it shows that the of SO(3) is of order 2. In physics applications, , and is an important tool in the development of the .}} Back to top Orientation entanglement :From Wikipedia:Orientation entanglement In three dimensions...the is not . Mathematically, one can tackle this problem by exhibiting the , which is also the in three dimensions, as a of SO(3). is the following group, : \mathrm{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbf{C}, |\alpha|^2 + |\beta|^2 = 1\right \} ~, where the overline denotes . For comparison: Using 2 × 2 complex matrices, the quaternion can be represented as : \begin{bmatrix} a+bi & c+di \\ -(c-di) & a-bi \end{bmatrix}. If X'' = (''x''1,''x''2,''x''3) is a vector in '''R'3, then we identify X'' with the 2 × 2 matrix with complex entries : X=\left(\begin{matrix}x_1&x_2-ix_3\\x_2+ix_3&-x_1\end{matrix}\right) Note that −det(''X) gives the square of the Euclidean length of X'' regarded as a vector, and that ''X is a , or better, trace-zero . The unitary group acts on X'' via : X\mapsto MXM^+ where ''M ∈ SU(2). Note that, since M'' is unitary, : \det(MXM^+) = \det(X) , and : MXM^+ is trace-zero Hermitian. Hence SU(2) acts via rotation on the vectors ''X. Conversely, since any which sends trace-zero Hermitian matrices to trace-zero Hermitian matrices must be unitary, it follows that every rotation also lifts to SU(2). However, each rotation is obtained from a pair of elements M'' and −''M of SU(2). Hence SU(2) is a double-cover of SO(3). Furthermore, SU(2) is easily seen to be itself simply connected by realizing it as the group of unit , a space to the . A unit quaternion has the cosine of half the rotation angle as its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis (here assumed fixed) as its pseudovector (or axial vector) part. If the initial orientation of a rigid body (with unentangled connections to its fixed surroundings) is identified with a unit quaternion having a zero pseudovector part and +1 for the scalar part, then after one complete rotation (2pi rad) the pseudovector part returns to zero and the scalar part has become -1 (entangled). After two complete rotations (4pi rad) the pseudovector part again returns to zero and the scalar part returns to +1 (unentangled), completing the cycle. Back to top Rotors From Wikipedia:Rotor A '''rotor is an object in (a ) that s any or general about the . They are normally motivated by considering an even number of s, which generate rotations (see also the ). The term originated with , in showing that the algebra is just a special case of 's "theory of extension" (Ausdehnungslehre). Hestenes defined a rotor to be any element R of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies \tilde{R} R = 1 , where \tilde{R} is the "reverse" of R —that is, the product of the same vectors, but in reverse order. Using inverse in geometric algebra may be represented as (minus) sandwiching a multivector ''M between a vector v'' perpendicular to the of reflection and that vector's ''v−1: : -v_1Mv_1^{-1} and are of even grade. Two reflections is a rotation. : v_2 v_1Mv_1^{-1} v_2^{-1} Multiplying a vector times a vector ( v_2 v_1 ) results in a bivector. When used in this manner we call the bivectors rotors. Under a rotation generated by the rotor R'', a general multivector ''M will transform double-sidedly as : RMR^{-1}. The formulation above (using inverse) is self-normalizing. Using reverse For a , it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of a unit (i.e. normalized) multivector: : -vMv, \quad v^2=1 , forming rotors that are automatically normalised: : RR^{\dagger}=R^{\dagger}R=1 . The derived rotor action is then expressed as a sandwich product with the reverse: : RMR^{\dagger} For a reflection for which the associated vector squares to a negative scalar, as may be the case with a , such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortcoming. Two rotations is given by : R_2R_1MR_1^{\dagger}R_2^{\dagger} = R_2R_1M(R_1R_2)^{\dagger} The alternative formulation above (using reverse) is not self-normalizing and motivates the definition of in geometric algebra as an object that transforms single-sidedly – i.e. spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product. Spinors :See also: External link:An introduction to spinors